Paper-19-20-2009. 117KB Jun 04 2011 12:03:11 AM

ACTA UNIVERSITATIS APULENSIS

No 20/2009

CONVERGENCE THEOREMS FOR UNIFORMLY L-LIPSCHITZIAN
ASYMPTOTICALLY NONEXPANSIVE MAPPINGS

Arif Rafiq
Abstract. Let K be a nonempty closed convex subset of a real Banach space E,
T : K → K a uniformly L-Lipschitzian asymptotically pseudocontractive mapping
with sequence {kn }n≥0 ⊂ [1, ∞), lim kn = 1 such that p ∈ F (T ) = {x ∈ K : T x =
n→∞

x}. Let {an }n≥0 , {bn }n≥0 , {cn }n≥0 beP
real sequences in [0, 1] satisfying the following
conditions: (i) an + bn + cn = 1; (ii) n≥0 bn = ∞; (iii) cn = o(bn ); (iv) lim bn = 0.
n→∞

For arbitrary x0 ∈ K let {xn }n≥0 be iteratively defined by

xn+1 = an xn + bn T n xn + cn un , n ≥ 0,

where {un }n≥0 is a bounded sequence of error terms in K. Suppose there exists a
strictly increasing function ψ : [0, ∞) → [0, ∞), ψ(0) = 0 such that
hT n x − p, j(x − p)i ≤ kn ||x − p||2 − ψ(||x − p||), ∀x ∈ K.
Then {xn }n≥0 converges strongly to p ∈ F (T ).
The results proved in this paper significantly improve the results of Ofoedu [11].
The remark 3 is important.
2000 Mathematics Subject Classification: Primary 47H10, 47H17: Secondary
54H25.
1.Introduction
Let E be a real normed space and K be a nonempty convex subset of E. Let J
∗
denote the normalized duality mapping from E to 2E defined by
J(x) = {f ∗ ∈ E ∗ : hx, f ∗ i = ||x||2 and ||f ∗ || = ||x||},
where E ∗ denotes the dual space of E and h·, ·i denotes the generalized duality
pairing. We shall denote the single-valued duality map by j.
Let T : D(T ) ⊂ E → E be a mapping with domain D(T ) in E.
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A. Rafiq - Convergence theorems for uniformly L-Lipschitzian asymptotically...

Definition 1 The mapping T is said to be uniformly L-Lipschitzian if there exists
L > 0 such that for all x, y ∈ D(T )
kT n x − T n yk ≤ L kx − yk .
Definition 2 T is said to be nonexpansive if for all x, y ∈ D(T ), the following
inequality holds:
kT x − T yk ≤ kx − y k for all x, y ∈ D(T ).
Definition 3 T is said to be asymptotically nonexpansive [6], if there exists a sequence {kn }n≥0 ⊂ [1, ∞) with lim kn = 1 such that
n→∞

kT n x − T n yk ≤ kn kx − y k for all x, y ∈ D(T ), n ≥ 1.
Definition 4 T is said to be asymptotically pseudocontractive if there exists a sequence {kn }n≥0 ⊂ [1, ∞) with lim kn = 1 and there exists j(x − y) ∈ J(x − y) such
n→∞
that
hT n x − T n y, j(x − y)i ≤ kn kx − y k2 for all x, y ∈ D(T ), n ≥ 1.
Remark 1 1. It is easy to see that every asymptotically nonexpansive mapping is
uniformly L-Lipschitzian.
2. If T is asymptotically nonexpansive mapping then for all x, y ∈ D(T ) there
exists j(x − y) ∈ J(x − y) such that
hT n x − T n y, j(x − y)i ≤ kT n x − T n yk kx − y k
≤ kn kx − y k2 , n ≥ 1.

Hence every asymptotically nonexpansive mapping is asymptotically pseudocontractive.
3. Rhoades in [12] showed that the class of asymptotically pseudocontractive
mappings properly contains the class of asymptotically nonexpansive mappings.
The asymptotically pseudocontractive mappings were introduced by Schu [13]
who proved the following theorem:
Theorem 1 Let K be a nonempty bounded closed convex subset of a Hilbert space H,
T : K → K a completely continuous, uniformly L-Lipschitzian and asymptotically
P
pseudocontractive with sequence {kn } ⊂ [1, ∞); qn = 2kn − 1, ∀n ∈ N ; (qn2 − 1) <
∞; {αn }, {β n } ⊂ [0, 1]; ǫ < αn < β n ≤ b, ∀n ∈ N, and some ǫ > 0 and some
1
b ∈ (0, L−2 [(1 + L2 ) 2 − 1]); x1 ∈ K for all n ∈ N, define
xn+1 = (1 − αn ) xn + αn T n xn .
Then {xn } converges to some fixed point of T .
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A. Rafiq - Convergence theorems for uniformly L-Lipschitzian asymptotically...

The recursion formula of theorem 1 is a modification of the well-known Mann
iteration process (see [9]).

Recently, Chang [1] extended Theorem 1 to real uniformly smooth Banach space;
in fact, he proved the following theorem:
Theorem 2 Let K be a nonempty bounded closed convex subset of a real uniformly
smooth Banach space E, T : K → K an asymptotically pseudocontractive mapping
with sequence {kn }n≥0 ⊂ [1, ∞), lim kn = 1, and x∗ ∈ F (T ) = {x ∈ K : T x = x}.
n→∞
P
Let {αn } ⊂ [0, 1] satisfying the following conditions: lim αn = 0,
αn = ∞. For
n→∞

arbitrary x0 ∈ K let {xn } be iteratively defined by

xn+1 = (1 − αn ) xn + αn T n xn , n ≥ 0.
If there exists a strictly increasing function ψ : [0, ∞) → [0, ∞), ψ(0) = 0 such that
hT n x − x∗ , j(x − x∗ )i ≤ kn ||x − x∗ ||2 − ψ(||x − x∗ ||), ∀n ∈ N,
then xn → x∗ ∈ F (T ).
Remark 2 Theorem 2, as stated is a modification of Theorem 2.4 of Chang [1] who
actually included error terms in his algorithm.
In [11], E. U. Ofoedu proved the following results.

Theorem 3 Let K be a nonempty closed convex subset of a real Banach space E,
T : K → K a uniformly L-Lipschitzian asymptotically pseudocontractive mapping
with sequence {kn }n≥0 ⊂ [1, ∞), lim kn = 1 such that x∗ ∈ F (T ) = {x ∈ K :
n→∞
P
P
2
α
=
∞,
T
x
=
x}.
Let
{α
}
⊂
[0,
1]

be
such
that
n
n
n≥0
n≥0 αn < ∞ and
n≥0
P
n≥0 αn (kn − 1) < ∞. For arbitrary x0 ∈ K let {xn }n≥0 be iteratively defined by
xn+1 = (1 − αn ) xn + αn T n xn , n ≥ 0.

Suppose there exists a strictly increasing function ψ : [0, ∞) → [0, ∞), ψ(0) = 0 such
that
hT n x − x∗ , j(x − x∗ )i ≤ kn ||x − x∗ ||2 − ψ(||x − x∗ ||), ∀x ∈ K.
Then {xn }n≥0 is bounded.
Theorem 4 Let K be a nonempty closed convex subset of a real Banach space E,
T : K → K a uniformly L-Lipschitzian asymptotically pseudocontractive mapping
with sequence {kn }n≥0 ⊂ [1, ∞), lim kn = 1 such that x∗ ∈ F (T ) = {x ∈ K :
n→∞

185

A. Rafiq - Convergence theorems for uniformly L-Lipschitzian asymptotically...

P
P
2
T
n≥0 αn = ∞,
n≥0 αn < ∞ and
Px = x}. Let {αn }n≥0 ⊂ [0, 1] be such that
n≥0 αn (kn − 1) < ∞. For arbitrary x0 ∈ K let {xn }n≥0 be iteratively defined by
xn+1 = (1 − αn ) xn + αn T n xn , n ≥ 0.

Suppose there exists a strictly increasing function ψ : [0, ∞) → [0, ∞), ψ(0) = 0 such
that
hT n x − x∗ , j(x − x∗ )i ≤ kn ||x − x∗ ||2 − ψ(||x − x∗ ||), ∀x ∈ K.
Then {xn }n≥0 converges strongly to x∗ ∈ F (T ).
Theorem 5 Let K be a nonempty closed convex subset of a real Banach space E,

T : K → K a uniformly L-Lipschitzian asymptotically pseudocontractive mapping
with sequence {kn }n≥0 ⊂ [1, ∞), lim kn = 1 such that x∗ ∈ F (T ) = {x ∈ K : T x =
n→∞

x}. Let {an }n≥0 , {bn }n≥0 , {cn }n≥0 be real sequences in [0, 1] satisfying the following
conditions:
i) an + bn + cn = 1;
P
ii)
n≥0 (bn + cn ) = ∞;
P
2
iii)
n≥0 (bn + cn ) < ∞;
P
iv)
n≥0 (bn + cn )(kn − 1) < ∞; and
P
v)
n≥0 cn < ∞.

For arbitrary x0 ∈ K let {xn }n≥0 be iteratively defined by
xn+1 = an xn + bn T n xn + cn un , n ≥ 0,

where {un }n≥0 is a bounded sequence of error terms in K. Suppose there exists a
strictly increasing function ψ : [0, ∞) → [0, ∞), ψ(0) = 0 such that
hT n x − x∗ , j(x − x∗ )i ≤ kn ||x − x∗ ||2 − ψ(||x − x∗ ||), ∀x ∈ K.
Then {xn }n≥0 converges strongly to x∗ ∈ F (T ).
Remark 3 P
One can easily P
see that if we take in theorems 3 and 4, αn = n1σ ; 0 <
σ < 1, then
αn = ∞, but
α2n = ∞. Hence the conclusions of theorems 3, 4 and
5 can be improved. The same argument can be applied on the results of [5].

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In this paper our purpose is to improve the results
a significantly
P of Ofoedu [11] inP
more general context by removing the conditions n≥0 α2n < ∞ and n≥0 αn (kn −
1) < ∞ from the theorems 3 − 4. We also significantly extend theorem 2 from
uniformly smooth Banach space to arbitrary real Banach space. The boundedness
assumption imposed on K in the theorem is also dispensed with.
2.Main Results
The following lemmas are now well known.
Lemma 6 [14] Let J : E → 2E be the normalized duality mapping. Then for any
x, y ∈ E, we have
||x + y||2 ≤ ||x||2 + 2hy, j(x + y)i, ∀j(x + y) ∈ J(x + y).
Suppose there exists a strictly increasing function ψ : [0, ∞) → [0, ∞) with ψ(0) = 0.
Lemma 7 [10] Let {θn } be a sequence of nonnegative real numbers, {λn } be a real
sequence satisfying
∞
X
0 ≤ λn ≤ 1,
λn = ∞
n=0

and let ψ ∈ Ψ. If there exists a positive integer n0 such that
θ2n+1 ≤ θ2n − λn ψ(θn+1 ) + σ n ,

for all n ≥ n0 , with σ n ≥ 0, ∀n ∈ N, and σ n = 0(λn ), then limn→∞ θn = 0.
Theorem 8 Let K be a nonempty closed convex subset of a real Banach space E,
T : K → K a uniformly L-Lipschitzian asymptotically pseudocontractive mapping
with sequence {kn }n≥0 ⊂ [1, ∞), lim kn = 1 such that p ∈ F (T ) = {x ∈ K : T x =
n→∞

x}. Let {an }n≥0 , {bn }n≥0 , {cn }n≥0 be real sequences in [0, 1] satisfying the following
conditions:
i) an + bn + cn = 1;
P
ii)
n≥0 bn = ∞;

iii) cn = o(bn );

iv) lim bn = 0.
n→∞

187

A. Rafiq - Convergence theorems for uniformly L-Lipschitzian asymptotically...

For arbitrary x0 ∈ K let {xn }n≥0 be iteratively defined by
xn+1 = an xn + bn T n xn + cn un , n ≥ 0,

(2.1)

where {un }n≥0 is a bounded sequence of error terms in K. Suppose there exists a
strictly increasing function ψ : [0, ∞) → [0, ∞), ψ(0) = 0 such that
hT n x − p, j(x − p)i ≤ kn ||x − p||2 − ψ(||x − p||), ∀x ∈ K.

(2.2)

Then {xn }n≥0 converges strongly to p ∈ F (T ).
Proof. From the condition cn = o(bn ) implies cn = tn bn , where tn → 0 as n → ∞.
Since p is a fixed point of T , then the set of fixed points F (T ) of T is nonempty.
Since the sequence {un }n≥0 is bounded, we set
M = sup kun − pk .
n≥0

By lim bn = 0 = lim tn imply there exists n0 ∈ N such that ∀n ≥ n0 , bn ≤ δ;
n→∞

n→∞

1
φ−1 (a0 )
,
,
18[φ−1 (a0 )]2 2(2 + L)φ−1 (a0 ) + M
)
φ(2φ−1 (a0 ))

 ,
12(1 + L)φ−1 (a0 ) 2(2 + L)φ−1 (a0 ) + M

0 < δ = min

and



tn ≤

φ(2φ−1 (a0 ))
.
12φ−1 (a0 )(M + 2φ−1 (a0 ))

Define a0 := kxn0 − T n0 xn0 kkxn0 − pk + (kn0 − 1)kxn0 − pk2 . Then from (2.2), we
obtain that kxn0 − pk ≤ φ−1 (a0 ).
CLAIM. kxn − pk ≤ 2φ−1 (a0 ) ∀n ≥ n0 .
The proof is by induction. Clearly, the claim holds for n = n0 . Suppose it holds
for some n ≥ n0 , i.e., kxn − pk ≤ 2φ−1 (a0 ). We prove that kxn+1 − pk ≤ 2φ−1 (a0 ).
Suppose that this is not true. Then kxn+1 −pk > 2φ−1 (a0 )), so that φ(kxn+1 −pk) >
φ(2φ−1 (a0 )). Using the recursion formula (2.1), we have the following estimates
kxn − T n xn k ≤ kxn − pk + kp − T n xn k
≤ (1 + L)kxn − pk
≤ 2(1 + L)φ−1 (a0 ),

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A. Rafiq - Convergence theorems for uniformly L-Lipschitzian asymptotically...

kxn+1 − pk = kan xn + bn T n xn + cn un − pk
= kxn − p − bn (xn − T n xn ) + cn (un − xn )k
≤ kxn − pk + bn kxn − T n xn k + cn kun − xn k
≤ 2φ−1 (a0 ) + 2(1 + L)φ−1 (a0 )bn + (M + 2φ−1 (a0 ))cn
≤ 2φ−1 (a0 ) + [2(2 + L)φ−1 (a0 ) + M ]bn
≤ 3φ−1 (a0 ).
With these estimates and again using the recursion formula (2.1), we obtain by
lemma 1 that
kxn+1 − pk2 = kan xn + bn T n xn + cn un − pk2
= kxn − p − bn (xn − T n xn ) + cn (un − xn )k
≤ kxn − pk2 − 2bn hxn − T n xn , j(xn+1 − p)i
+2cn hun − xn , j(xn+1 − p)i
= kxn − pk2 + 2bn hT n xn+1 − p, j(xn+1 − p)i
−2bn hxn+1 − p, j(xn+1 − p)i
+2bn hT n xn − T n xn+1 , j(xn+1 − p)i
+2bn hxn+1 − xn , j(xn+1 − p)i
+2cn hun − xn , j(xn+1 − p)i


≤ kxn − pk2 + 2bn kn ||xn+1 − p||2 − φ(||xn+1 − p||)

−2bn kxn+1 − pk2 + 2bn kT n xn − T n xn+1 k ||xn+1 − p||
+2bn kxn+1 − xn k ||xn+1 − p||
+2cn (M + kxn − pk) kxn+1 − pk

≤ kxn − pk2 + 2bn (kn − 1)||xn+1 − p||2 − 2bn φ(||xn+1 − p||)
+2bn (1 + L) kxn+1 − xn k ||xn+1 − p||
+2cn (M + kxn − pk) kxn+1 − pk ,

(2.3)

where
kxn+1 − xn k = kan xn + bn T n xn + cn un − xn k
= kbn (T n xn − xn ) + cn (un − xn )k
≤ bn kxn − T n xn k + cn kun − xn k
≤ 2(1 + L)φ−1 (a0 )bn + (M + 2φ−1 (a0 ))cn
≤ [2(2 + L)φ−1 (a0 ) + M ]bn .
189

(2.4)

A. Rafiq - Convergence theorems for uniformly L-Lipschitzian asymptotically...

Substituting (2.4) in (2.3), we get
kxn+1 − pk2 ≤ kxn − pk2 − 2bn φ(2φ−1 (a0 ))
+18[φ−1 (a0 )]2 (kn − 1)bn
+6(1 + L)φ−1 (a0 )[2(2 + L)φ−1 (a0 ) + M ]b2n
+6φ−1 (a0 )(M + 2φ−1 (a0 ))cn
≤ kxn − pk2 + (kn − 1) − φ(2φ−1 (a0 ))bn .
Thus
φ(2φ−1 (a0 ))bn ≤ kxn − pk2 − kxn+1 − pk2 + (kn − 1),
implies

φ(2φ−1 (a0 ))

j
X

bn ≤

n=n0

j
X

(kxn − pk2 − kxn+1 − pk2 ) +

n=n0

j
X

(kn − 1)

n=n0

= kxn0 − pk2 +

j
X

(kn − 1),

n=n0

so that as j → ∞ we have
φ(2φ−1 (a0 ))

∞
X

bn ≤ kxn0 − pk2 +

n=n0

j
X

(kn − 1) < ∞,

n=n0

P
which implies that
bn < ∞, a contradiction. Hence, kxn+1 − x∗ k ≤ 2φ−1 (a0 );
thus {xn } is bounded.
Now with the help (2.4) and the condition cn = o(bn ), (2.3) takes the form
kxn+1 − pk2

≤ ||xn − p||2 − 2bn φ(kxn+1 − pk)
+2bn [4[φ−1 (a0 )]2 (kn − 1)
+2(1 + L)φ−1 (a0 )[2(2 + L)φ−1 (a0 ) + M ]bn
+2φ−1 (a0 )(M + 2φ−1 (a0 ))tn ].

Denote
θn = ||xn − p||,
λn = 2bn ,
σ n = 2bn [4[φ−1 (a0 )]2 (kn − 1)
+2(1 + L)φ−1 (a0 )[2(2 + L)φ−1 (a0 ) + M ]bn
+2φ−1 (a0 )(M + 2φ−1 (a0 ))tn ].
190

(2.5)

A. Rafiq - Convergence theorems for uniformly L-Lipschitzian asymptotically...

Condition lim bn = 0 assures the existence of a rank n0 ∈ N such that λn = 2bn ≤ 1,
n→∞
P
for all n ≥ n0 . Now with the help of n≥0 bn = ∞, cn = o(bn ), lim bn = 0 and
n→∞

Lemma 2, we obtain from (2.5) that

lim ||xn − p|| = 0,

n→∞

completing the proof.
Corollary 9 Let K be a nonempty closed convex subset of a real Banach space E,
T : K → K a uniformly L-Lipschitzian asymptotically pseudocontractive mapping
with sequence {kn }n≥0 ⊂ [1, ∞), lim kn = 1 such that p ∈ F (T ) = {x ∈ K : T x =
n→∞P
x}. Let {αn }n≥0 ⊂ [0, 1] be such that n≥0 αn = ∞ and lim αn = 0. For arbitrary
n→∞

x0 ∈ K let {xn }n≥0 be iteratively defined by

xn+1 = (1 − αn ) xn + αn T n xn , n ≥ 0.
Suppose there exists a strictly increasing function ψ : [0, ∞) → [0, ∞), ψ(0) = 0 such
that
hT n x − p, j(x − p)i ≤ kn ||x − p||2 − ψ(||x − p||), ∀x ∈ K.
Then {xn }n≥0 converges strongly to p ∈ F (T ).

References
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asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 129 (2000) 845–
853.
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their generalizations, Nonlinear Anal. (to V. Lakshmikantham on his 80th birthday)
1,2 (2003) 383–429.
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Lipschitz generalized phi-quasiaccretive operators, Proc. Amer. Math. Soc., 134
(2006) 243-251.
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303 (2005) 545–554.

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[6] K. Goebel,W. A. Kirk, A fixed point theorem for asymptotically nonexpansive
mappings, Proc. Amer. Math. Soc. 35 (1972) 171–174.
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Arif Rafiq
Dept. of Mathematics,
COMSATS Institute of Information Technology,
Lahore, Pakistan
email: arafiq@comsats.edu.pk

192